Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad.)

Then cover $M$ by open sets $\cup_iU_i=M$. In a local coordinate chart, $(U_i,\phi_i)$, where $\phi_i:U_i\rightarrow \mathbb{R}^n$, let us denote these local coordinates by $(\sigma^1,\dots,\sigma^n)$.

**My question is:** *under what conditions do coordinates such as $(\sigma^1,\dots,\sigma^n)$ exist that cover the entire manifold, $M$, and more importantly why?*

Related questions are: what is the obstruction to extending the local chart to cover the entire surface (except possibly for a discrete set of points in $M$)? Is there a general reasoning that applies to all cases (at least for the case of an orientable compact Riemann surface)?

Any help much appreciated!

suspectthat it's equivalent to manifold being affine (i. e. factor of $\Bbb R^k$ by properly discontinuous action of $Aff(k, \Bbb R)$. $\endgroup$ – Denis T. Aug 22 '18 at 19:56holomorphiccoordinates. Any ideas why? thanks! $\endgroup$ – Wakabaloola Aug 22 '18 at 20:53